Nonparametric approaches to estimation problems for demand andsupply functions in power exchange markets

Yuji Yamada

Abstract— In this paper, we demonstrate two approaches toestimate supply and demand functions in the Japan ElectricPower Exchange (JEPX) spot market based on the sell andbuy matching rates (SMRs and BMRs) and the nonparametricsimultaneous equations. In the first approach, we apply thegeneralized additive model (GAM) of the contract price on theSMR and the BMR to derive the volume rate functions, wherethe SMR and the BMR are defined using the contract volumedivided by the total selling and buying volumes. We demonstratethat supply and demand functions are obtained by performingthe coordinate transformations for the volume rate functions.In the nonparametric simultaneous equations approach, we usethe total selling and buying volumes as exogenous variables andshow that supply and demand functions may be constructedbased on parametric representations using GAMs. An empiricalanalysis is also included to compare the two approaches.

I. INTRODUCTION

The role of electricity spot market is to provide anopportunity to meet supply and demand effectively throughthe ask and bid orders. In the case of Japan Electric PowerExchange (JEPX) spot market, the spot contract is made oneday before delivering the electricity, and all the ask and bidorders are closed at a specified time of each day. Then,the JEPX constructs supply and demand curves in whichthe ask (sell) and bid (buy) orders volumes with acceptableprices are accumulated, and the intersection of supply anddemand curves is computed to determine the contract priceand volume. Although the supply and demand curves providequite important information regarding the price-volume rela-tionship for suppliers and demanders in electricity market,one can usually observe partial information such as thecontract price and price given by the intersection of supplyand demand curves1. The objective of this paper is to developa methodology to estimate supply and demand functions forthe JEPX based on the available observed data2.

The difficulty for estimating supply and demand functionsis that we cannot identify two functions from the single pointinformation, say the intersection given by the contract priceand volume (see e.g., [1]). In this paper, we demonstrate twoapproaches to estimate supply and demand functions in theJEPX spot market based on the sell and buy matching rates(SMRs and BMRs [11]) and the nonparametric simultaneousequations. At first, we define the SMR and the BMR using

Faculty of Business Sciences, University of Tsukuba, Tokyo, Japan. Thiswork is supported by Grant-in-Aid for Scientific Research (B) 25282087from Japan Society for the Promotion of Science (JSPS).

1In the case of JEPX, the contract price and volume data is available aswell as the total volumes for selling and buying orders.

2Note that it has become important to analyze the structure of JEPX spotmarket both empirically and theoretically (see, e.g., [3], [4], [6], [7], [10]).

the contract volume divided by the total selling and the totalbuying volumes, respectively. Then, we apply a nonparamet-ric regression called the generalized additive model (GAM[2]) for the contract price on the SMR and the BMR andderive volume rate functions. We show that the price-volumefunctions corresponding to supply and demand functions areobtained by using the coordinate transformations.

Next, we develop the nonparametric simultaneous equa-tions approach, where the total selling and buying volumesare used as exogenous variables. We show that supply anddemand functions may be constructed based on parametricrepresentations using GAMs, in which we extend the stan-dard simultaneous equations approach to a nonparametriccase. Since the supply and demand functions in the elec-tricity market should have nonlinearity due to the nonlineardependence of energy cost on the volume, one can expect abetter fit by using nonparametric regressions.

Finally, we compare the two approaches using empiricaldata. We discuss that the estimation results from the matchingrates approach are more consistent in terms of t-valuesregarding the monotonicity of supply and demand functions,although the GAMs for nonparametric simultaneous equa-tions approach may have higher contribution rates to explainprice fluctuations than that for the matching rates approach.

II. ESTIMATION OF SUPPLY AND DEMAND FUNCTIONSAND MATCHING-RATES APPROACH

A. Problem formulation

Let P (t)n and V

(t)n be the contract price (JPY/MWh) and

the contract volume (MWh) in the JEPX spot market beingdelivered at day n (= 1, . . . , N) during time t (= 0, . . . , 23)to t+1 for 1 hour. In the JEPX, the contract price and volume,P

(t)n and V

(t)n , are usually determined one day before (i.e.,

day n − 1) for each t = 0, . . . , 23 by constructing supplyand demand functions as shown in the left hand side of Fig.1, where the upward-sloping (or the downward-sloping) linedescribes the relationship between the unit price (JPY/MWh)and the sellable (or buyable) volume (MWh) offered bysuppliers (or demanders). In the figure, the intersection of thetwo lines provides

(P

(t)n , V

(t)n

), which may be regarded as

the equilibrium price and volume of JEPX spot market. Theobjective of this study is to estimate the supply and demandfunctions for given period index (t, n) using observed data.

One possible idea for estimating supply or demand func-tion is to regress P (t)

n with respect to V (t)n for different values

of market data(P

(t)n , V

(t)n

)with t fixed as

Pn = βVn + c+ ϵn, n = 1, . . . , N (1)

2015 IEEE 54th Annual Conference on Decision and Control (CDC)December 15-18, 2015. Osaka, Japan

978-1-4799-7885-4/15/$31.00 ©2015 IEEE 1781

Fig. 1. Volume functions (left) and volume rate functions (right)

where time index t is omitted for brevity, and β, c and ϵnare a regression coefficient, a constant term and a residual,respectively. Although β in (1) is expected to be positivefor supply functions and negative for demand functions,these two parameters cannot be identified from the singleregression equation (1) only. Moreover, since the cost ofelectricity (e.g., JPY/MWh) generated by different sources orplants may vary significantly and have nonlinear dependenceon the volume, we may need to apply non-parametric regres-sion techniques to take nonlinearity of supply and demandfunctions into consideration. In this paper, we discuss twoapproaches to estimate supply and demand functions. Thefirst is based on the sell and buy matching rates (SMRs andBMRs) given below, and the second approach is aimed togeneralize the simultaneous equations using linear regres-sions for a nonparametric case as explained in Section III.

B. Sell and buy matching-rates approach

Let VS,n and VB,n be the total selling and buying volumesat day n (with time index t although being omitted), which infact, denote the maximum sellable and the maximum buyablevolumes. Note that VS,n and VB,n provide the maximumvalues of volume domains. If the bid and ask prices areavailable, VS,n and VB,n correspond to the x-coordinates ofthe highest bid price and the lowest ask price in the supplyand demand functions, respectively.

Let the sell and buy matching rates be denoted by Sn :=Vn/VS,n and Bn := Vn/VB,n. The relation between the con-tract price Pn and Sn (or Bn) is depicted using a coordinatetransformation of supply (or demand) function as shown inthe right hand side of Fig. 1, where the upward-sloping (orthe downward-sloping) line is obtained by changing the scaleof x-coordinate as x/VS,n (or x/VB,n) of the supply (or thedemand) function. In this case, x/VS,n (or x/VB,n) may beconsidered as the rate of volume per total and we refer tothese functions as selling and buying volume rate functions.On the other hand, the original ones are said to be sellingand buying volume functions in this paper.

In contrast to equation (1) for selling and buying volumefunctions, one can distinguish sell and buy matching ratesgiven contract price Pn, and selling and buying volume rate

functions may be estimated based on the following equations:

Pn = fs (Sn) + ϵS,n, Pn = gb (Bn) + ϵB,n (2)

where fs and gs are smooth functions, and ϵS,n andϵB,n, n = 1, . . . , N residuals. In this case, the selling andbuying volume functions on the (x, y)-coordinate plane(providing supply and demand functions) may be constructedby multiplying VS,n or VB,n with the x-coordinate.

Let fs, gb, ϵS,n and ϵB,n be estimators of the smoothfunctions and the residuals. Then, supply and demand func-tions on (x, y)-coordinate plane may be constructed as

y = fs(x/VS,n

)+ ϵS,n, y = gb

(x/VB,n

)+ ϵB,n. (3)

Note that, the functions, fs and gb, have to be monotonicfunctions of upward-sloping or downward-sloping, i.e.,

∂fs∂x

> 0,∂gb∂x

< 0, (4)

and that the functions in (3) intersect at (x, y) = (Vn, Pn)from the definitions of Sn and Bn.

It is worthwhile to mention that supply and demandfunctions may be estimated simultaneously based on thefollowing single regression equation:

Pn = fs (Sn) + gb (Bn) + ϵn, n = 1, . . . , N. (5)

where fs and gb are smooth functions and are estimatedusing GAMs. Let estimators of the smooth functions and theresidual be given by fs = fs, gb = gb, and ϵn. Then, supplyand demand functions on (x, y)-coordinate plane may beconstructed as follows:

y = fs(x/VS,n

)+ gb (Bn) + ϵn,

y = gb(x/VB,n

)+ fs (Sn) + ϵn.

(6)

We see that the supply and demand functions in (6) corre-spond to those in (3) by letting

ϵS,n ≡ gb (Bn) + ϵn, ϵB,n ≡ gb (Sn) + ϵn.

One can regard as if two regression equations of (2) areexpressed using a single equation, (5). That is, equation (5)is interpreted that the JEPX spot price, Pn, is regressed withrespect to a supply variable, Sn, controlled by a demandvariable Bn, but at the same time, that Pn is regressed withrespect to Sn controlled by Bn.

For applying GAM (5), we can also take calendar andtemperature effects into account using temperature, a longterm trend, and day and holiday dummy variables. Let Tnstand for the temperature at day n weighted by the percentageof population in each region. Also let Monn, . . . , Satn andHolidayn be dummy variables such that, e.g., Monn = 1 ifday n is Monday or Monn = 0 otherwise (and so on). Then,the GAM used in our analysis is formulated as follows:

Pn = fs (Sn) + gb (Bn) + u (Tn) + β1Monn + · · ·+ β6Satn + β7Holidayn + β8n+ εn (7)

where u is a smooth function of Tn, and βi, i = 1, . . . , 8and εn are regression coefficients and a residual.

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III. NONPARAMETRIC SIMULTANEOUS EQUATIONSAPPROACH

The simultaneous equations approach is based on theidea that different coordinate points of one function may beobserved by shifting the other using exogenous variables. Tobegin with, let Z1,n and Z2,n be exogenous variables andconsider the following equations:

Pn = fv (Vn + hz1 (Z1,n)) + kz1 (Z1,n) + ϵf,n, (8)Pn = gv (Vn + hz2 (Z2,n)) + kz2 (Z2,n) + ϵg,n (9)

where fv, gv, hz1, kz1, hz2, and kz2 are smooth functions,and ϵf,n and ϵg,n are residuals. The above equations are in-troduced to derive supply and demand functions using fv andgv , where the exogenous variables Z1,n and Z2,n are chosento shift (x, y)-coordinates of supply and demand functions,respectively. Although equations (8) and (9) provide thestandard linear simultaneous equations model if fv, . . . , kz2are given by linear functions, direct estimation of (8) or (9)using (standard or nonstandard) regression techniques maybe difficult in the case of general nonparametric functions.Also, there is a fundamental problem characterized as thesimultaneous equations bias so that equations (8) and (9) can-not be estimated directly by applying separate regressions.

Here we show that the following equations provide para-metric representations of supply and demand functions in thenonparametric simultaneous equations approach:

Pn = ϕz1 (Z1,n) + ϕz2 (Z2,n) + εϕ,n, (10)Vn = ψz1 (Z1,n) + ψz2 (Z2,n) + εψ,n, (11)

where ϕz1, ϕz2, ψz1, and ψz2 are smooth functions to beestimated using GAMs, and εϕ,n and εψ,n are residuals. Letϕz1, ϕz2, ψz1, and ψz2 be estimators of smooth functions,and Pn and Vn predicted values of Pn and Vn, i.e.,

Pn = ϕz1 (Z1,n) + ϕz2 (Z2,n) = Pn − εϕ,n, (12)

Vn = ψz1 (Z1,n) + ψz2 (Z2,n) = Vn − εψ,n. (13)

By solving equation (13) with respect to Z1,n and Z2,n andsubstituting them into equation (12) lead to the followings:

Pn = fv

(Vn + hz1 (Z1,n)

)+ kz1 (Z1,n) , (14)

Pn = gv

(Vn + hz2 (Z2,n)

)+ kz2 (Z2,n) , (15)

where the smooth functions, ψz1, and ψz2, are assumed tobe invertible, and fv ≡ ϕz2 ◦ ψ−1

z2 , hz1 ≡ −ψz1, kz1 ≡ ϕz1,gv ≡ ϕz1 ◦ ψ−1

z1 , hz2 ≡ −ψz2 and kz2 ≡ ϕz2. We seethat equation (14) and (15) provide estimators of equations,(8) and (9), using predicted values of Pn and Vn. Then,the supply and demand functions intersecting at (x, y)= (Vn, Pn) are, respectively, represented on the (x, y)-coordinate plane as

y = fv

(x+ hz1 (Z1,n)

)+ kz1 (Z1,n) , (16)

y = gv

(x+ hz2 (Z2,n)

)+ kz2 (Z2,n) . (17)

Note that one can express the functions (16) and (17) basedon parametric representations using GAMs. To confirm this,let us replace one of the exogenous variables, Z2,n in (12)and (13), with a real parameter z2, and introduce a parametricrepresentation of (x, y)-coordinate point (xs, ys) as{

xs = ψz1 (Z1,n) + ψz2 (z2) ,

ys = ϕz1 (Z1,n) + ϕz2 (z2) .(18)

We see that equations in (18) provide a parametric functionof z2, which is actually equivalent to the explicit functionalrepresentation of supply function (16). Similarly, by replac-ing Z1,n with z1 in (12) and (13), we obtain the followingparametric representation, (xd, yd), of demand function (17):{

xd = ψz1 (z1) + ψz2 (Z2,n) ,

yd = ϕz1 (z1) + ϕz2 (Z2,n) .(19)

When z1 = Z1,n and z2 = Z2,n, it holds that (xs, ys) =(xd, yd) =

(Vn, Pn

).

Moreover, the supply and demand functions may bedefined using residuals, εϕ,n and εψ,n, whose parametricrepresentations are modified as{

xs = ψz1 (Z1,n) + ψz2 (z2) + εψ,n,

ys = ϕz1 (Z1,n) + ϕz2 (z2) + εϕ,n,(20){

xd = ψz1 (z1) + ψz2 (Z2,n) + εψ,n,

yd = ϕz1 (z1) + ϕz2 (Z2,n) + εϕ,n.(21)

In this case, it is readily verified that the supply and demandfunctions of (20) and (21) intersect at the coordinate pointof realized values for contract volume and price, (Vn, Pn).

Now, we discuss the monotonic conditions for supply anddemand functions. Since the supply and demand functionsare monotonic increasing and decreasing, we have that f ′v >0 and g′v < 0 hold, or equivalently,

∂ys/∂z2∂xs/∂z2

=∂ϕz2/∂z2

∂ψz2/∂z2> 0,

∂yd/∂z1∂xd/∂z1

=∂ϕz1/∂z1

∂ψz1/∂z1< 0.

hold in the parametric representation. Noting that ψz1 andψz2 are assumed to be invertible and so are monotonic,potential candidates of exogenous variables are to chooseZ1,n ≡ VS,n and Z2,n ≡ VB,n due to the following reasons:

• if the total selling volume, VS,n, is increased, thecontract volume is expected to increase but the pricemay decrease due to a high supply, and

• if the total buying volume, VB,n, is increased, both thecontract volume and price are expected to increase dueto a high demand.

The above conditions implies that ϕz1 and ψz1 are, respec-tively, expected to be monotonic decreasing and increasingfunctions, whereas ϕz2 and ψz2 both monotonic increasingfunctions, i.e.,

∂ϕz1∂z1

< 0,∂ψz1∂z1

> 0,∂ϕz2∂z2

> 0,∂ψz2∂z2

> 0. (22)

Similar to the regression equation (7) for matching ratesapproach, we can apply GAMs by taking calendar and

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temperature effects into account for constructing the non-parametric simultaneous equations as follows:

Pn = hz1(VS,n

)+ hz2

(VB,t

)+ up (Tn) + β1pMonn + · · ·

+ β6pSatn + β7pHolidayn + β8pn+ εp,n (23)Vn = kz1

(VS,n

)+ kz2

(VB,t

)+ uv (Tn) + β1vMonn + · · ·

+ β6vSatn + β7vHolidayn + β8vn+ εv,n (24)

where up and uv are smooth functions of Tn, and βip andβiv, i = 1, . . . , 8 regression coefficients.

IV. MONOTONIC TRANSFORMATION

As stated in Section II, supply and demand functionsmust be monotonic, implying that the smooth functions, fsand gb, in (7) are monotonic, as well as those in (23) and(24) for the nonparametric simultaneous equations approach.However, the estimators of smooth functions in GAMs arenot necessarily monotonic, and as a result, the estimatedsupply and demand functions may not be monotonic, either.In this section, we show that the smooth functions estimatedin GAMs can be transformed to monotonic functions byapplying quadratic optimization problems.

Assume that we have obtained a smooth function f s.t.

yn = f (xn) + c+ ϵn, Mean [f (xn)] = 0 (25)

for given sample points, (xn, yn), where c is a constant term,ϵn a residual satisfying Mean [ϵn] = 0, and Mean [·] standsfor sample mean. Without loss of generality, let x1 ≤ x2 ≤· · · ≤ xN , and consider the following optimization problem:

minz1,...,zN

N∑n=1

[f (xn)− zn]2 (26)

s.t. z1 ≤ z2 ≤ · · · ≤ zN

z1 + · · ·+ zN = 0

The problem (26) is a linear constrained quadratic opti-mization and may be solved using, e.g., an interior pointmethod to find optimizers zn, n = 1, . . . , N . Let f bea function s.t. f(xn) = zn, n = 1, . . . , N . Then, linearinterpolation may be applied so that f is a monotonicincreasing function satisfying

f (x1) ≤ f (x2) ≤ · · · ≤ f (xN ) , Mean[f (xn)

]= 0.

We see that f minimizes the mean square error betweenf (xn) and zn with monotonicity condition on the samplepoints. Although f may not be smooth, Wolberg and Alfy [9]have proposed an interpolation technique for cubic splines,which should be investigated in our future research.

Let R2 be the coefficient of determination of the originalregression equation (25), and let ϵn = yn − f (xn). Definethe coefficient of determination after the monotonic transfor-mation as

R2 := 1− Variance [ϵn] /Variance [yn] , (27)

where Variance[·] is sample variance. Clearly, it holds thatR2 ≤ R2, and the larger the difference, the larger the

discrepancy of the original and the monotonic transformedfunctions, f and f . Therefore, we can estimate the effect ofmonotonic transformation by comparing those coefficients ofdetermination, R2 and R2.

V. EMPIRICAL ANALYSIS

In this section, we demonstrate our proposed method-ologies for estimating supply and demand functions usingempirical data for the JEPX spot price, contract volume,total selling and buying volumes, and temperature in theperiod from August 8, 2005, to September 30, 2014 (n =1, . . . , N, N = 3, 341)3.

Time (hour)0 5 10 15 20

R2 fo

r mon

oton

ic fu

nctio

n

0.5

0.6

0.7

0.8R2 for monotonic function and the distance from original

Diff

eren

ce o

f R2 (o

rigin

al-m

onot

ocic

)

0

0.005

0.01

0.015

Distance from original

0

0.005

0.01

0.015

Time (hour)0 5 10 15 20

T-va

lue

-50

-40

-30

-20

-10

0

10

20T-values of coefficients

Sell rateBuy rate5% significance level

Fig. 2. Coefficients of determination for the matching rates approach(upper) and t-values of supply and demand functions (lower)

At first, we apply GAM (7) to estimate supply and demandfunctions based on the matching rates approach and investi-gate the effect of monotonic transformation. The solid line inthe upper plot of Fig. 2 the coefficient of determinations, R2,after the monotonic transformations for each t = 0, . . . , 23with the y-axis on the left, where the tips of the verticallines are those of R2 before the monotonic transformations.The difference, R2 − R2, is also denoted by the dashed linewith the y-axis on the right. From this result, we first seethat the effect of monotonic transformation on the coefficientof determination is very small, say at most less than 1.5%.

3Note that there are missing values before these periods in spite thatJEPX spot trading started in the beginning of April 2005. The data maybe downloaded from the JEPX website (http://www.jepx.org/).Also, the temperature data is observed every 24 hours at each dayand may be downloaded from the Japan Meteorological Agency website(http://www.jma.go.jp/).

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Moreover, since the coefficients of determination providecontribution rates of selling and buying matching rates (andthe other control variables), we see that about 60–70% ofprice fluctuations may be explained by explanatory variablesusing the regression equation of GAM (7).

To examine the monotonic properties, we compute the t-values of coefficients related to supply and demand functionsby replacing the spline functions of selling and buyingmatching rates, fs (Sn) and gb (Bn) in GAM (7), withβsSn and βbSn. Note that the coefficients βs and βb maybe considered as the average increase and decrease rates,respectively. For example, if the t-value of βs is positivelysignificant. the price Pn increases on average, and we caninterpret that fs is expected to be a monotonic increasingfunction on average. Similarly, gs may be regarded as amonotonic decreasing function on average if the t-value ofβb is negatively significant.

The black and gray lines in the lower plot of Fig. 2,respectively, provide the t-values for Sn and Bn (i.e., βsand βb in the above explanation) with respect to time t =0, . . . , 23, where the dashed lines are 5% significant levels.We see that the black line is always above the 5% significantlevel and that the coefficients of Sn are positively significant,implying that the supply functions obtained from matchingrates approach are monotonic increasing on average. Also,the gray line is always below the 5% significant level,implying that the demand functions from matching ratesapproach are monotonic decreasing on average.

Time (hour)0 5 10 15 20

R2 fo

r mon

oton

ic fu

nctio

n

0.6

0.7

0.8

0.9

1R2 for monotonic function and the distance from original

Distance from original

0

0.02

0.04

Diff

eren

ce o

f R2 (o

rigin

al-m

onot

ocic

)

0

0.01

0.02

0.03

0.04

Time (hour)0 5 10 15 20

R2 fo

r mon

oton

ic fu

nctio

n

0.8

0.85

0.9

0.95R2 for monotonic function and the distance from original

Diff

eren

ce o

f R2 (o

rigin

al-m

onot

ocic

)

×10-3

-1

0

1

2

Distance from original

×10-3

-1

0

1

2

Fig. 3. Coefficients of determination in GAMs (23) (upper) and (24) (lower)for the nonparametric simultaneous equations approach

Next, we apply GAMs (23) and (24) for estimating supply

and demand functions based on the nonparametric simulta-neous equations approach. Similar to the upper plot of Fig. 2,we compare the coefficients of determination before and afterthe monotonic transformations as shown in Fig. 3, where theupper and lower plots show our results for GAMs (23) and(24), respectively.

We see that the coefficients of determination correspond-ing to the contribution rates are about 10% higher than thosefor GAM (7) for explaining the price fluctuations using thetotal selling and buying volumes instead of matching rates.Also, the coefficients of determination for explaining thevolume fluctuations are also high using GAM (24). However,the difference of coefficients of determination before andafter monotonic transformations are slightly larger than thosefor GAM (7), which indicates that the effect of monotonictransformation may be larger.

To evaluate the monotonicity, we replace ϕz1(VS,n

),

ϕz2(VB,n

), ψz1

(VS,n

), and ψz2

(VB,n

)in (23) and (24)

by βz1VS,n, βz2VS,n, γz1VS,n and γz2VS,n, and compute t-values, similar to those in the lower plot of Fig. 2 for match-ing rates approach. Recalling that the monotonic conditionsfor the supply and demand functions are given by (22), weexpect that βz1 < 0 and βz2, γz1, γz2 > 0, in which it isclearer to depict the t-values whose signs are expected to bepositive together in one plot and negative in the other.

Time (hour)0 5 10 15 20

T-va

lue

0

10

20

30

40

50

60

70T-values of coefficients

Sell in volumeBuy in volumeBuy in price5% significance level

Time (hour)0 5 10 15 20

T-va

lue

-12

-10

-8

-6

-4

-2

0

2T-values of coefficients

Sell in price5% significance level

Fig. 4. t-values of supply and demand functions for the nonparametricsimultaneous equations approach

The upper plot in Fig. 4 shows the t-values for coefficientsβz2, γz1 and γz2 whose signs are expected to be positive,and the lower that for βz1 having expected negative sign.The black, the dark gray, and the gray lines in the upper

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plot, respectively, show the t-values for βz2, γz1, γz2, and theblack in the lower that for βz1. Noting that the dashed lines inboth plots denote the 5% significance level, we see that βz2,γz1, γz20 are all positively significant. On the other hand,we can verify that βz1 is not always negatively significant.In particular, there are some cases that βz1 is significant (orclose to significant) with the opposite sign indicating thatthe monotonic decreasing property for demand function maynot hold. In summary, we conclude that, although GAM (23)for the nonparametric simultaneous equations approach mayhave higher contribution rates to explain price fluctuationsthan that for the matching rates approach, the estimation re-sults from the nonparametric simultaneous equations may beinconsistent in terms of t-values regarding the monotonicityof demand functions.

Finally, we demonstrate supply and demand functionsestimated from the two approaches, where we choose thedate when the prices of 9am take its median only (whichis on May-2-2007) during the entire data period due to thespace constraint. The black and gray lines in Fig. 5 showsthe estimated supply and demand functions, respectively,where the upper plot is for the matching rates approachand the lower for the nonparametric simultaneous equationsapproach. At first, we see that the supply and demandfunctions intersect the coordinate of realized values for thecontract volume and price, (Vn, Pn), where the dashed linesindicate x = Vn and y = Pn.

Volume [MWh]100 200 300 400 500 600 700 800

Pow

er p

rice

[Yen

/kW

h]

10

15

20

25

30

02-May-2007

SellBuyExecuted

Volume [MWh]100 200 300 400 500 600 700 800

Pow

er p

rice

[Yen

/kW

h]

10

15

20

25

30

02-May-2007

SellBuyExecuted

Fig. 5. Supply and demand functions of 9am price on May-2-2007 for thematching rates (upper) and the simultaneous equations (lower) approaches

Also, we realize that the shapes of them are a littledifferent between the matching rates and the nonparametric

simultaneous equations approaches. For example, the rangesof functions for the matching rates approach are wider in thex-axis directions than those for the nonparametric simulta-neous equations approach. On the other hand, in the y-axisdirections, the ranges for the nonparametric simultaneousequations approach are wider than those for the matchingrates approach. Noting that the maximum and the minimumprices for ask and bid are, in fact, unopened, the y-rangesof supply and demand functions are undetermined. For thismatter, it should be interesting to investigate how to fit (orapproximate) the ranges of y-axis in our future research.

VI. CONCLUDING REMARKS

In this paper, we have considered estimation problems forsupply and demand functions in the JEPX spot markets,where we have developed two approaches based on thesell and buy matching rates (SMRs and BMRs) and thenonparametric simultaneous equations. In the first approach,we applied GAMs for the contract price on the SMR andBMR to estimate volume rate functions, and then, con-structed the supply and demand functions using coordinatetransformations. In the second approach, we extended thestandard simultaneous equations approach for a nonparamet-ric case, where we proposed to apply GAMs to estimate thenonparametric simultaneous equations to derive parametricrepresentations of supply and demand functions. Finally,we compared the two approaches using empirical data andconcluded that, although the GAMs for nonparametric si-multaneous equations approach may have higher contributionrates to explain price fluctuations than that for the matchingrates approach, the estimation results from the matching ratesapproach are more consistent in terms of t-values regardingthe monotonicity of supply and demand functions.

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